- #1
rohanprabhu
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[SOLVED] Surprising Gradient not 'Surprising Enough'
Q] Sketch the vector function and
<br /> v = \frac{\hat{r}}{r^2}<br />
and compute it's divergence. The answer may surprise you... can you explain it?
['r' is the position vector in the Euclidean space]
<br /> \nabla \cdot v = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}<br />
<br /> \mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}<br />
and,
<br /> \mathbf{\hat{r}} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2 + y^2 + z^2}}<br />
Hence,
<br /> \frac{\mathbf{\hat{r}}}{r^2} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}<br />
<br /> \nabla \cdot v = \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3x^2)}{(x^2 + y^2 + z^2)^3} + \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3y^2)}{(x^2 + y^2 + z^2)^3} + \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3z^2)}{(x^2 + y^2 + z^2)^3}<br />
on using Simplify in Mathematica:
<br /> \nabla \cdot v = \frac{3(-1 + x^4 + y^4 + 2y^2 z^2 + z^4 + 2x^2 y^2 + 2x^2 z^2))}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}<br />
But there was nothing 'surprising' about this result. What am I doing it wrong?
Also, this problem is from D. Griffith's 'Introduction to Electrodynamics'. And I am at the 2nd chapter which deals with Calculus used in Electrodynamics. The thing is that I am heavily confused between a Gradient, Divergence and Curl. Hence, to overcome that, I think the only way is practicing a lot of questions. Could anyone please show me any source where I could get lots of questions on Gradient, Divergence and Curl to practice [it'd be nice if it were a free resource].
thanks
Homework Statement
Q] Sketch the vector function and
<br /> v = \frac{\hat{r}}{r^2}<br />
and compute it's divergence. The answer may surprise you... can you explain it?
['r' is the position vector in the Euclidean space]
Homework Equations
<br /> \nabla \cdot v = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}<br />
The Attempt at a Solution
<br /> \mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}<br />
and,
<br /> \mathbf{\hat{r}} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2 + y^2 + z^2}}<br />
Hence,
<br /> \frac{\mathbf{\hat{r}}}{r^2} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}<br />
<br /> \nabla \cdot v = \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3x^2)}{(x^2 + y^2 + z^2)^3} + \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3y^2)}{(x^2 + y^2 + z^2)^3} + \frac{(x^2 + y^2 + z^2)^{\frac{1}{2}}((x^2 + y^2 + z^2)^3 - 3z^2)}{(x^2 + y^2 + z^2)^3}<br />
on using Simplify in Mathematica:
<br /> \nabla \cdot v = \frac{3(-1 + x^4 + y^4 + 2y^2 z^2 + z^4 + 2x^2 y^2 + 2x^2 z^2))}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}<br />
But there was nothing 'surprising' about this result. What am I doing it wrong?
Also, this problem is from D. Griffith's 'Introduction to Electrodynamics'. And I am at the 2nd chapter which deals with Calculus used in Electrodynamics. The thing is that I am heavily confused between a Gradient, Divergence and Curl. Hence, to overcome that, I think the only way is practicing a lot of questions. Could anyone please show me any source where I could get lots of questions on Gradient, Divergence and Curl to practice [it'd be nice if it were a free resource].
thanks
